Setting Targets is All You Need:Improved Order Competitive Ratio for Online Selection

TL;DR

This paper introduces targeted value algorithms for online selection, achieving improved order-competitive ratios and answering whether randomized algorithms can outperform deterministic ones in this setting.

Contribution

The paper presents a novel family of algorithms that set targeted values, achieving optimal and near-optimal order-competitive ratios, and demonstrates that online selection is as easy as guessing the optimal benchmark.

Findings

Deterministic targeted value algorithm achieves a 1/φ order-competitive ratio.

Randomized targeted value algorithm achieves a 0.732 ratio.

Upper bounds of 0.758 and 0.829 on the ratio for specific algorithms.

Abstract

There is a rising interest for studying the online benchmark as an alternative of the classical offline benchmark in online stochastic settings. Ezra, Feldman, Gravin, and Tang (SODA 2023) introduced the notion of order-competitive ratio, defined as the worst-case ratio between the performance of the best order-unaware algorithm and the best order-aware algorithm, to quantify the loss incurred by the lack of knowledge of the arrival order. They showed in the online single selection setting (a.k.a. the prophet problem), the optimal order-competitive ratio achieved by deterministic algorithms is $1/\varphi \approx 0.618$, and left with an open question whether randomized algorithms can do better. We answer the open question firmly by introducing a novel family of algorithms called \emph{targeted value algorithms}. We show that the task of online selection is as easy as guessing the optimal online benchmark. Specifically, we provide 1) an alternative optimal $1/\varphi$ order-competitive algorithm by setting the targeted value deterministically, and 2) a $0.732$ order-competitive algorithm by setting the targeted value randomly. We further provide a $0.758$ upper bound on the order-competitive ratio of our algorithm, showing that our analysis is close to the best possible, and establish an upper bound of $0.829$ on the order-competitive ratio for general randomized order-unaware algorithms.